This is a Phase I/II study (a study that examines effectiveness and looks for side effects). The transplanting of islet cells has been studied in Type 1 diabetic patients whose blood sugar levels will not stay normal, despite intensive insulin therapy. A recent study conducted in Edmonton, Canada, was able to demonstrate that islet transplantation led to insulin independence in a majority of the patients treated. This study extends the results obtained from the Edmonton study, which used islet transplantation in Type 1 diabetic patients with steroid-free immunosuppression.